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An Estimation of the Gradient Norm of Biharmonic Functions
DOI: 10.12677/PM.2023.1310296, PDF, HTML, XML, 下载: 134  浏览: 1,756

Abstract: Biharmonic functions play an important role in the field of mathematics and have wide applications in reality. This article mainly explores an estimate of the gradient norm of biharmonic functions. By analyzing the relationship between biharmonic functions and bianalytic functions, the Poisson kernel of biharmonic functions is calculated, and an estimate of the gradient norm of bounded biharmonic functions is given. The obtained integral expression lays the foundation for further exploration of the Khavinson conjecture of biharmonic functions in the future.

1. 基础知识

${\Delta }^{2}\varnothing \left(r,\theta \right)=\Delta \Delta \varnothing \left(r,\theta \right)=\frac{{\partial }^{4}\varnothing \left(r,\theta \right)}{\partial {x}^{4}}+2\frac{{\partial }^{4}\varnothing \left(r,\theta \right)}{\partial {x}^{2}\partial {y}^{2}}+\frac{{\partial }^{4}\varnothing \left(r,\theta \right)}{\partial {y}^{4}}=0$ (1)

$\forall 0\le r\le 1,0\le \theta <2\pi$

0在极坐标下，对于单位圆盘内的双调和函数：

$\varnothing \left(r,\theta \right)=\left({r}^{2}-1\right){U}_{1}\left(r,\theta \right)+{U}_{2}\left(r,\theta \right)$ (2)

${\varnothing \left(r,\theta \right)|}_{r=1}=\mu \left(\theta \right)$ (3)

${\varnothing \left(r,\theta \right)|}_{r=1}={\left({1}^{2}-1\right){U}_{1}\left(r,\theta \right)+{U}_{2}\left(r,\theta \right)|}_{r=1}={{U}_{2}\left(\theta \right)|}_{r=1}=\mu \left(\theta \right)$ (4)

${U}_{2}\left(r,\theta \right)=\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{\left(1-{r}^{2}\right)\mu \left(\psi \right)}{1-2r\mathrm{cos}\left(\theta -\psi \right)+{r}^{2}}\text{d}\psi$ (5)

${\frac{\partial \varnothing }{\partial r}|}_{r=1}=0$ (6)

${\frac{\partial \varnothing \left(r,\theta \right)}{\partial r}|}_{r=1}={\left[2{U}_{1}\left(r,\theta \right)+\frac{\partial {U}_{2}\left(r,\theta \right)}{\partial r}\right]|}_{r=1}=0$ (7)

$\begin{array}{c}{\frac{\partial {U}_{2}\left(r,\theta \right)}{\partial r}|}_{r=1}=\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{-2\mu \left(\psi \right)\left[-2\mathrm{cos}\left(\theta -\psi \right)+2r\right]}{{\left[-2\mathrm{cos}\left(\theta -\psi \right)+2r\right]}^{2}}\text{d}\psi \\ =-\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{\mu \left(\psi \right)}{1-\mathrm{cos}\left(\theta -\psi \right)}\text{d}\psi \end{array}$

${{U}_{1}\left(r,\theta \right)|}_{r=1}=\frac{1}{4\text{π}}{\int }_{0}^{2\text{π}}\frac{\mu \left(\psi \right)}{1-\mathrm{cos}\left(\theta -\psi \right)}\text{d}\psi$ (8)

$\begin{array}{c}{U}_{1}\left(r,\theta \right)=\frac{1}{4\text{π}}{\int }_{0}^{2\text{π}}\frac{\frac{1-{r}^{2}}{2\text{π}}{\int }_{0}^{2\text{π}}\frac{\mu \left(\psi \right)}{1-\mathrm{cos}\left(\omega -\psi \right)}\text{d}\psi }{1-2r\mathrm{cos}\left(\theta -\psi \right)+{r}^{2}}\text{d}\omega \\ =\frac{1-{r}^{2}}{8{\text{π}}^{2}}{\int }_{0}^{2\text{π}}\mu \left(\psi \right)\text{d}\psi {\int }_{0}^{2\text{π}}\frac{1}{\left(1-\mathrm{cos}\left(\omega -\psi \right)\right)\left[1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}\right]}\text{d}\omega \end{array}$

$\begin{array}{l}U\left(z\right)=-\frac{{\left({r}^{2}-1\right)}^{2}}{8{\text{π}}^{2}}{\int }_{0}^{2\text{π}}\mu \left(\psi \right)\text{d}\psi {\int }_{0}^{2\text{π}}\frac{1}{\left[1-\mathrm{cos}\left(\omega -\psi \right)\right]\left[1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}\right]}\text{d}\omega \\ \text{}+\frac{1-{r}^{2}}{2\text{π}}{\int }_{0}^{2\text{π}}\frac{\mu \left(\psi \right)}{1-2r\mathrm{cos}\left(\theta -\psi \right)+{r}^{2}}\text{d}\psi ,\text{}\forall 0\le r\le 1,0\le \theta <2\text{π}\end{array}$ (9)

$P\left(z,\zeta \right)=-\frac{{\left({r}^{2}-1\right)}^{2}}{4\text{π}}{\int }_{0}^{2\text{π}}\frac{1}{\left[1-\mathrm{cos}\left(\omega -\psi \right)\right]\left[1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}\right]}\text{d}\omega +\frac{1-{r}^{2}}{{|z-\zeta |}^{2}},0\le r\le 1,0\le \theta <2\text{π}$ (10)

$U\left(z\right)$ 是单位圆盘D上的有界双调和函数，则对几乎所有的 $\zeta \in \partial D$ ，都有径向边界值 [4] ：

${U}^{*}\left(\zeta \right)={\mathrm{lim}}_{r\to {1}^{-}}U\left(r\zeta \right)$ (11)

$U\left(z\right)$ 可以用 ${U}^{*}\left(\zeta \right)$ 的Poisson积分表示

$U\left(z\right)=P\left[{U}^{*}\right]\left(z\right)={\int }_{\partial D}P\left(z,{U}^{*}\left(\zeta \right)\right){U}^{*}\left({U}^{*}\left(\zeta \right)\right)\text{d}s\left({U}^{*}\left(\zeta \right)\right)$ (12)

$P\left(z,\zeta \right)=-\frac{{\left({r}^{2}-1\right)}^{2}}{4\text{π}}{\int }_{0}^{2\text{π}}\frac{1}{\left[1-\mathrm{cos}\left(\omega -\psi \right)\right]\left[1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}\right]}\text{d}\omega +\frac{1-{r}^{2}}{{|z-\zeta |}^{2}}$ (13)

$〈\nabla U\left(z\right),l〉={\int }_{\partial D}〈\nabla P\left(z,\zeta \right),l〉{U}^{*}\left(\zeta \right)\text{d}s\left(\zeta \right)$ . (14)

$‖\Lambda ‖=C\left(z,l\right)$ , (15)

$|〈\nabla u,l〉|\le C\left(z,l\right){\mathrm{sup}}_{w\in D}|u\left(w\right)|$

$C\left(z,l\right)={\int }_{\partial D}|〈\nabla P\left(z,\zeta \right),l〉|\text{d}s\left(\zeta \right)$ (16)

2. C(z, l)的积分表达式

$\begin{array}{l}{\int }_{0}^{2\pi }\frac{1}{\left[1-\mathrm{cos}\left(\omega -\psi \right)\right]\left[1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}\right]}\text{d}\omega \\ =\frac{2\pi r\mathrm{cos}\left(\psi -\theta \right)\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]-4\pi {r}^{2}{\mathrm{sin}}^{2}\left(\psi -\theta \right)}{{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{2}\left(1-{r}^{2}\right)}\end{array}$ (17)

$\begin{array}{c}\frac{\partial }{\partial \psi }{\int }_{0}^{2\pi }\frac{\mathrm{cot}\frac{\omega -\psi }{2}}{1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}}\text{d}\omega ={\int }_{0}^{2\pi }\frac{\frac{\partial }{\partial \psi }\mathrm{cot}\frac{\omega -\psi }{2}}{1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}}\text{d}\omega \\ ={\int }_{0}^{2\pi }\frac{1}{\left[1-\mathrm{cos}\left(\omega -\psi \right)\right]\left[1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}\right]}\text{d}\omega \end{array}$ (18)

$t=-\mathrm{cot}\frac{\omega -\psi }{2}$ ，得

$\frac{\mathrm{cot}\frac{\omega -\psi }{2}\text{d}\omega }{1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}}=\frac{t\text{d}t}{1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}-4rt\mathrm{sin}\left(\psi -\theta \right)+\left(1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right){t}^{2}}$

${\int }_{0}^{2\pi }\frac{\mathrm{cot}\frac{\omega -\psi }{2}}{1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}}\text{d}\omega =\frac{2\pi r\mathrm{sin}\left(\psi -\theta \right)}{\left(1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right)\left(1-{r}^{2}\right)}$ (19)

$\begin{array}{l}\frac{\partial }{\partial \psi }{\int }_{0}^{2\pi }\frac{\mathrm{cot}\frac{\omega -\psi }{2}}{1-2r\mathrm{cos}\left(\theta -\omega \right)+{r}^{2}}\text{d}\omega \\ =\frac{2\pi r\mathrm{cos}\left(\psi -\theta \right)\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]-4\pi {r}^{2}{\mathrm{sin}}^{2}\left(\psi -\theta \right)}{{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{2}\left(1-{r}^{2}\right)}\end{array}$

$\begin{array}{c}P\left(r,\theta \right)=-\frac{1-{r}^{2}}{2}\left\{\frac{r\mathrm{cos}\left(\psi -\theta \right)\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]-2{r}^{2}{\mathrm{sin}}^{2}\left(\psi -\theta \right)}{{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{2}}\right\}\\ \text{\hspace{0.17em}}\text{ }+\frac{1-{r}^{2}}{1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}}\end{array}$ (20)

$\begin{array}{c}〈\nabla P\left(z,\zeta \right),l〉=\frac{-3{r}^{3}{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}\mathrm{cos}\theta +{r}^{5}\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}-\frac{4r\mathrm{cos}\theta {\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}{4{|z-\zeta |}^{6}}+\frac{27r\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}+\frac{17{r}^{3}\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)-5\left[\frac{{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}{r}\right]\mathrm{cos}\theta }{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}+\frac{-40{r}^{3}\mathrm{cos}\theta +3\left(\frac{1+{r}^{2}-{|z-\zeta |}^{2}}{r}\right)\mathrm{cos}\theta -4r\mathrm{cos}\theta }{4{|z-\zeta |}^{6}}\end{array}$

$\begin{array}{l}+\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left[5\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)\right]}{4{|z-\zeta |}^{6}}\\ +\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left[{r}^{2}\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)\right]}{4{|z-\zeta |}^{6}}\\ +\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left({r}^{4}-10{r}^{2}-3\right)}{4{|z-\zeta |}^{6}}\\ 0\le r\le 1,0\le \theta <2\pi \end{array}$ (21)

$\begin{array}{c}P\left(r,\theta \right)=-\frac{1-{r}^{2}}{2}\left\{\frac{r\mathrm{cos}\left(\psi -\theta \right)\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]-2{r}^{2}{\mathrm{sin}}^{2}\left(\psi -\theta \right)}{{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{2}}\right\}\\ \text{\hspace{0.17em}}+\frac{1-{r}^{2}}{1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}}\end{array}$

$\left\{\begin{array}{c}\frac{\partial P}{\partial x}=\frac{\partial P}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial P}{\partial \theta }\frac{\partial \theta }{\partial x}①\\ \frac{\partial P}{\partial y}=\frac{\partial P}{\partial r}\frac{\partial r}{\partial y}+\frac{\partial P}{\partial \theta }\frac{\partial \theta }{\partial y}②\end{array}$

$\frac{\partial P}{\partial \theta }$$\frac{\partial P}{\partial r}$ 求解：

$\frac{\partial P}{\partial \theta }=\frac{\left({r}^{2}-1\right)r\mathrm{sin}\left(\psi -\theta \right)\left[2{r}^{3}\mathrm{cos}\left(\psi -\theta \right)+{r}^{4}+10r\mathrm{cos}\left(\psi -\theta \right)-10{r}^{2}-3\right]}{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}$ (22)

$\begin{array}{c}\frac{\partial P}{\partial r}=\frac{27{r}^{2}\mathrm{cos}\left(\psi -\theta \right)-20{r}^{3}+3\mathrm{cos}\left(\psi -\theta \right)-4r}{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}+\frac{17{r}^{4}\mathrm{cos}\left(\psi -\theta \right)-10r{\mathrm{cos}}^{2}\left(\psi -\theta \right)}{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}+\frac{-6{r}^{5}{\mathrm{cos}}^{2}\left(\psi -\theta \right)+{r}^{6}\mathrm{cos}\left(\psi -\theta \right)-8{r}^{3}{\mathrm{cos}}^{2}\left(\psi -\theta \right)}{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\end{array}$ (23)

$\frac{\partial P}{\partial x}=\frac{\partial P}{\partial r}\mathrm{cos}\theta +\frac{\partial P}{\partial \theta }\left(-\frac{\mathrm{sin}\theta }{r}\right)=\frac{\partial P}{\partial r}\mathrm{cos}\theta -\frac{\partial P}{\partial \theta }\left(\frac{\mathrm{sin}\theta }{r}\right)$ (24)

$\frac{\partial P}{\partial y}=\frac{\partial P}{\partial r}\frac{\partial r}{\partial y}+\frac{\partial P}{\partial \theta }\frac{\partial \theta }{\partial y}=\frac{\partial P}{\partial r}\mathrm{sin}\theta +\frac{\partial P}{\partial \theta }\frac{\mathrm{cos}\theta }{r}$ (25)

$\begin{array}{c}\frac{\partial P}{\partial x}=-\frac{\left({r}^{2}-1\right)\mathrm{sin}\left(\psi -\theta \right)\left[2{r}^{3}\mathrm{cos}\left(\psi -\theta \right)+{r}^{4}+10r\mathrm{cos}\left(\psi -\theta \right)-10{r}^{2}-3\right]}{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}+\frac{-6{r}^{5}{\mathrm{cos}}^{2}\left(\psi -\theta \right)\mathrm{cos}\theta +{r}^{6}\mathrm{cos}\theta \mathrm{cos}\left(\psi -\theta \right)-8{r}^{3}{\mathrm{cos}}^{2}\left(\psi -\theta \right)\mathrm{cos}\theta }{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}+\frac{17{r}^{4}\mathrm{cos}\left(\psi -\theta \right)\mathrm{cos}\theta }{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\frac{-10r{\mathrm{cos}}^{2}\left(\psi -\theta \right)\mathrm{cos}\theta +27{r}^{2}\mathrm{cos}\left(\psi -\theta \right)\mathrm{cos}\theta }{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}-\frac{\left({r}^{2}-1\right)\mathrm{sin}\theta \mathrm{sin}\left(\psi -\theta \right)\left[2{r}^{3}\mathrm{cos}\left(\psi -\theta \right)+{r}^{4}+10r\mathrm{cos}\left(\psi -\theta \right)-10{r}^{2}-3\right]}{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}+\frac{-20{r}^{3}\mathrm{cos}\theta +3\mathrm{cos}\left(\psi -\theta \right)\mathrm{cos}\theta -4r\mathrm{cos}\theta }{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\end{array}$ (26)

$\begin{array}{c}\frac{\partial P}{\partial y}=\frac{\partial P}{\partial r}\mathrm{sin}\theta +\frac{\partial P}{\partial \theta }\frac{\mathrm{cos}\theta }{r}\\ =\frac{-6{r}^{5}{\mathrm{cos}}^{2}\left(\psi -\theta \right)\mathrm{sin}\theta +{r}^{6}\mathrm{cos}\left(\psi -\theta \right)\mathrm{sin}\theta -8{r}^{3}{\mathrm{cos}}^{2}\left(\psi -\theta \right)\mathrm{sin}\theta }{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}+\frac{17{r}^{4}\mathrm{cos}\left(\psi -\theta \right)\mathrm{sin}\theta }{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}+\frac{-10r{\mathrm{cos}}^{2}\left(\psi -\theta \right)\mathrm{sin}\theta +27{r}^{2}\mathrm{cos}\left(\psi -\theta \right)\mathrm{sin}\theta }{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}+\frac{\left({r}^{2}-1\right)\mathrm{cos}\theta \mathrm{sin}\left(\psi -\theta \right)\left[2{r}^{3}\mathrm{cos}\left(\psi -\theta \right)+{r}^{4}+10r\mathrm{cos}\left(\psi -\theta \right)-10{r}^{2}-3\right]}{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\\ \text{\hspace{0.17em}}+\frac{-20{r}^{3}\mathrm{sin}\theta +3\mathrm{cos}\left(\psi -\theta \right)\mathrm{sin}\theta -4r\mathrm{sin}\theta }{2{\left[1-2r\mathrm{cos}\left(\psi -\theta \right)+{r}^{2}\right]}^{3}}\end{array}$ (27)

${|z-\zeta |}^{2}=1-2r\mathrm{cos}\left(\theta -\psi \right)+{r}^{2}$ (28)

$\mathrm{cos}\left(\theta -\psi \right)=\frac{1+{r}^{2}-{|z-\zeta |}^{2}}{2r}$ (29)

${\mathrm{cos}}^{2}\left(\theta -\psi \right)=\frac{{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}{4{r}^{2}}$ (30)

$\mathrm{sin}\left(\theta -\psi \right)=\sqrt{1-{\mathrm{cos}}^{2}\left(\theta -\psi \right)}=\sqrt{1-\frac{{\left(1-{|z-\zeta |}^{2}+{r}^{2}\right)}^{2}}{4{r}^{2}}}$ (31)

$\nabla P\left(z,\zeta \right)=\left(\frac{\partial P}{\partial x},\frac{\partial P}{\partial y}\right)$ (32)

$\begin{array}{c}〈\nabla P\left(z,\zeta \right),l〉=\frac{-3{r}^{3}{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}\mathrm{cos}\theta +{r}^{5}\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}-\frac{4r\mathrm{cos}\theta {\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}{4{|z-\zeta |}^{6}}+\frac{27r\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}+\frac{17{r}^{3}\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)-5\left[\frac{{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}{r}\right]\mathrm{cos}\theta }{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}+{\frac{-40{r}^{3}\mathrm{cos}\theta +3\left(\frac{1+{r}^{2}-{|z-\zeta |}^{2}}{r}\right)\mathrm{cos}\theta -4r\mathrm{cos}\theta }{4{|z-\zeta |}^{6}}}_{{}_{}}\end{array}$

$\begin{array}{l}+\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left[5\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)\right]}{4{|z-\zeta |}^{6}}\\ +\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left[{r}^{2}\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)\right]}{4{|z-\zeta |}^{6}}\\ +\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left({r}^{4}-10{r}^{2}-3\right)}{4{|z-\zeta |}^{6}}\end{array}$ (33)

$\begin{array}{c}C\left(z,l\right)={\int }_{\partial D}\frac{17{r}^{3}\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)-5\left[\frac{{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}{r}\right]\mathrm{cos}\theta }{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}\frac{-3{r}^{3}{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}\mathrm{cos}\theta +{r}^{5}\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}-\frac{4r\mathrm{cos}\theta {\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}{4{|z-\zeta |}^{6}}+\frac{27r\mathrm{cos}\theta \left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}{4{|z-\zeta |}^{6}}\\ \text{\hspace{0.17em}}+\frac{-40{r}^{3}\mathrm{cos}\theta +3\left(\frac{1+{r}^{2}-{|z-\zeta |}^{2}}{r}\right)\mathrm{cos}\theta -4r\mathrm{cos}\theta }{4{|z-\zeta |}^{6}}\end{array}$

$\begin{array}{l}+\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left[5\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)\right]}{4{|z-\zeta |}^{6}}\\ +\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left[{r}^{2}\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)\right]}{4{|z-\zeta |}^{6}}\\ +\frac{\frac{{r}^{2}-1}{r}\mathrm{sin}\theta \sqrt{4{r}^{2}-{\left(1+{r}^{2}-{|z-\zeta |}^{2}\right)}^{2}}\left({r}^{4}-10{r}^{2}-3\right)}{4{|z-\zeta |}^{6}}\end{array}$ (34)

3. 结论

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